Concept guide
Linear maps: proving v↦Mv is linear, and Mv+b is linear exactly when b=0
For every 2×2 real matrix M, prove that T(v)=Mv is a linear map ℝ²→ℝ² (additivity and homogeneity for all vectors u, v and real c); prove that the affine map F(v)=Mv+b is linear exactly when b=0. For the example M₀=[[2,−1],[3,4]], b=(1,−2)ᵀ, u=(1,1)ᵀ, w=(2,−3)ᵀ: compute M₀u, show that F is not linear with an explicit failing equation, and illustrate additivity of T numerically; finally use g(x,y)=(x²,y) to show that finitely many test points cannot prove that an arbitrary map is linear.